On the number of conjugacy classes of normalisers in a finite $p$-group

In 1996 Poland and Rhemtulla proved that the number ν(G) of conjugacy classes of non-normal subgroups of a non-Hamiltonian nilpotent group G is at least c − 1, where c is the nilpotency class of G. In this paper we consider the map that associates to every conjugacy class of subgroups of a finite p-group the conjugacy class of the normalizer of any of its representatives. In spite of the fact that this map needs not to be injective, we prove that, for p odd, the number of conjugacy classes of normalizers in a finite p-group is at least c (taking into accout the normalizer of the normal sugroups). In the case of p-groups of maximal class we can find a better lower bound that depends also on the prime p.